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Stabilization for the semilinear waveequation with geometric control condition
Romain Joly∗ & Camille Laurent†‡
July 31, 2018
Abstract
In this article, we prove the exponential stabilization of the semilinear wave equa-tion with a damping effective in a zone satisfying the geometric control conditiononly. The nonlinearity is assumed to be subcritical, defocusing and analytic. Themain novelty compared to previous results, is the proof of a unique continuationresult in large time for some undamped equation. The idea is to use an asymptoticsmoothing effect proved by Hale and Raugel in the context of dynamical systems.Then, once the analyticity in time is proved, we apply a unique continuation resultwith partial analyticity due to Robbiano, Zuily, Tataru and Hormander. Some otherconsequences are also given for the controllability and the existence of a compactattractor.
Key words: damped wave equation, stabilization, analyticity, unique continuationproperty, compact attractor.AMS subject classification: 35B40, 35B60, 35B65, 35L71, 93D20, 35B41.
Resume
Dans cet article, on prouve la decroissance exponentielle de l’equation des on-des semilineaires avec un amortissement actif dans une zone satisfaisant seulementla condition de controle geometrique. La nonlinearite est supposee sous-critique,defocalisante et analytique. La principale nouveaute par rapport aux resultats pre-cedents est la preuve d’un resultat de prolongement unique en grand temps pourune solution non amortie. L’idee est d’utiliser un effet regularisant asymptotique
∗Institut Fourier - UMR5582 CNRS/Universite de Grenoble - 100, rue des Maths - BP 74 - F-38402St-Martin-d’Heres, France, email: [email protected]
†CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France‡UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, email:
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prouve par Hale et Raugel dans le contexte des systemes dynamiques. Ensuite, unefois l’analyticite en temps prouvee, on applique un theoreme de prolongement uniqueavec analyticite partielle du a Robbiano, Zuily, Tataru et Hormander. Des applica-tions a la controlabilite et a l’existence d’attracteur global compact pour l’equationdes ondes sont aussi donnees.
1 Introduction
In this article, we consider the semilinear damped wave equation
u+ γ(x)∂tu+ βu+ f(u) = 0 (t, x) ∈ R+ × Ω ,u(t, x) = 0 (t, x) ∈ R+ × ∂Ω(u, ∂tu) = (u0, u1) ∈ H1
0 (Ω)× L2(Ω)(1.1)
where = ∂2tt − ∆ with ∆ being the Laplace-Beltrami operator with Dirichlet boundary
conditions. The domain Ω is a connected C∞ three-dimensional Riemannian manifold withboundaries, which is either:
i) compact.
ii) a compact perturbation of R3, that is R3 \ D where D is a bounded smooth domain,
endowed with a smooth metric equal to the euclidean one outside of a ball.
iii) or a manifold with periodic geometry (cylinder, R3 with periodic metric etc.).
The nonlinearity f ∈ C1(R,R) is assumed to be defocusing, energy subcritical and suchthat 0 is an equilibrium point. More precisely, we assume that there exists C > 0 suchthat
f(0) = 0 , sf(s) ≥ 0 , |f(s)| ≤ C(1 + |s|)p and |f ′(s)| ≤ C(1 + |s|)p−1 (1.2)
with 1 ≤ p < 5.We assume β ≥ 0 to be such that ∆− β is a negative definite operator, that is that we
have a Poincare inequality∫Ω|∇u|2 + β|u|2 ≥ C
∫Ω|u|2 with C > 0. In particular, it may
require β > 0 if ∂Ω = ∅ or if Ω is unbounded.The damping γ ∈ L∞(Ω) is a non-negative function. We assume that there exist an
open set ω ⊂ Ω, α ∈ R, x0 ∈ Ω and R ≥ 0 such that
∀x ∈ ω , γ(x) ≥ α > 0 and Ω \B(x0, R) ⊂ ω . (1.3)
Moreover, we assume that ω satisfies the geometric control condition introduced in [44]and [7]
2
(GCC) There exists L > 0 such that any generalized geodesic of Ω of length L meets the setω where the damping is effective.
The associated energy E ∈ C0(X,R+) is given by
E(u) := E(u, ∂tu) =1
2
∫
Ω
(|∂tu|2 + |∇u|2 + β|u|2) +∫
Ω
V (u) , (1.4)
where V (u) =∫ u
0f(s)ds. Due to Assumption (1.2) and the Sobolev embedding H1(Ω) →
L6(Ω), this energy is well defined and moreover, if u solves (1.1), we have, at least formally,
∂tE(u(t)) = −∫
Ω
γ(x)|∂tu(x, t)|2 dx ≤ 0 . (1.5)
The system is therefore dissipative. We are interested in the exponential decay of theenergy of the nonlinear damped wave equation (1.1), that is the following property:
(ED) For any E0 ≥ 0, there exist K > 0 and λ > 0 such that, for all solutions u of (1.1)with E(u(0)) ≤ E0,
∀t ≥ 0 , E(u(t)) ≤ Ke−λtE(u(0))
Property (ED) means that the damping term γ∂tu stabilizes any solution of (1.1) to zero,which is an important property from the dynamical and control points of view.
Our main theorem is as follows.
Theorem 1.1. Assume that the damping γ satisfies (1.3) and the geometric control con-dition (GCC). If f is real analytic and satisfies (1.2), then the exponential decay property(ED) holds.
Theorem 1.1 applies for nonlinearities f which are globally analytic. Of course, thenonlinearities f(u) = |u|p−1u are not analytic if p 6∈ 1, 3, but we can replace these usualnonlinearities by similar ones as f(u) = (u/th(u))p−1u, which are analytic for all p ∈ [1, 5).Note that the estimates (1.2) are only required for s ∈ R, so that it does not imply that f ispolynomial. Moreover, we enhance that (ED) holds in fact for almost all the nonlinearitiesf satisfying (1.2), including non-analytic ones.
More precisely, we set
C1(R) = f ∈ C1(R) such that there exist C > 0 and p ∈ [1, 5) such that (1.2) holds
(1.6)endowed with Whitney topology (or any other reasonable topology). We recall that Whit-ney topology is the topology generated by the neighbourhoods
Nf,δ = g ∈ C1(R) | ∀u ∈ R , max(|f(u)− g(u)|, |f ′(u)− g′(u)|) < δ(u) (1.7)
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where f is any function in C1(R) and δ is any positive continuous function. The set C1(R)
is a Baire space, which means that any generic set, that is any set containing a countableintersection of open and dense sets, is dense in C
1(R) (see Proposition 7.1). Baire propertyensures that the genericity of a set in C
1(R) is a good notion for “the set contains almostall non-linearity f”.
Theorem 1.2. Assume that the damping γ satisfies (1.3) and the geometric control condi-tion (GCC). There exists a generic set G ⊂ C
1(R) such that the exponential decay property(ED) holds for all f ∈ G.
The statements of both theorems lead to some remarks.• Of course, our results and their proofs should easily extend to any space dimension d ≥ 3if the exponent p of the nonlinearity satisfies p < (d+ 2)/(d− 2).• Actually, it may be possible to get λ > 0 in (ED) uniform with respect to the size of thedata. We can take for instance λ = λ− ε where λ is the decay rate of the linear equation.The idea is that once we know the existence of a decay rate, we know that the solutionis close to zero for a large time. Then, for small solutions, the nonlinear term can beneglected to get almost the same decay rate as the linear equation. We refer for instanceto [36] in the context of KdV equation. Notice that the possibility to get the same resultwith a constant K independent on E0 is an open problem.• The assumption on β is important to ensure some coercivity of the energy and to precludethe spatially constant functions to be undamped solutions for the linear equation. It hasbeen proved in [14] for R3 and in [38] for a compact manifold that exponential decay canfail without this term β.• The geometric control condition is known to be not only sufficient but also necessary forthe exponential decay of the linear damped equation. The proof of the optimality uses somesequences of solutions which are asymptotically concentrated outside of the damping region.We can use the same idea in our nonlinear stabilization context. First, the observability fora certain time eventually large is known to be equivalent to the exponential decay of theenergy. This was for instance noticed in [14] Proposition 2, in a similar context, see alsoProposition 2.5 of this paper. Then, we take as initial data the same sequence that wouldgive a counterexample for the linear observability. The linearizability property (see [19])allows to obtain that the nonlinear solution is asymptotically close to the linear one. Thiscontradicts the observability property for the nonlinear solution as it does for the linearcase. Hence, the geometric control condition is also necessary for the exponential decay ofthe nonlinear equation.• Our geometrical hypotheses on Ω may look strange, however they are only assumed forsake of simplicity. In fact, our results should apply more generally for any smooth manifoldwith bounded geometry, that is that Ω can be covered by a set of C∞−charts αi : Ui 7−→αi(Ui) ⊂ R
3 such that αi(Ui) is equal either to B(0, 1) or to B+(0, 1) = x ∈ B(0, 1), x1 >0 (in the case with boundaries) and such that, for any r ≥ 0 and s ∈ [1,∞], theW r,s−normof a function u in W r,s(Ω,R) is equivalent to the norm (
∑i∈N ‖u α−1
i ‖sW r,s(αi(Ui)))1/s.
4
The stabilization property (ED) for Equation (1.1) has been studied in [28], [53], [54]and [13] for p < 3. For p ∈ [3, 5), our main reference is the work of Dehman, Lebeauand Zuazua [17]. This work is mainly concerned with the stabilization problem previouslydescribed, on the Euclidean space R
3 with flat metric and stabilization active outside of aball. The main purpose of this paper is to extend their result to a non flat geometry wheremultiplier methods cannot be used or do not give the optimal result with respect to thegeometry. Other stabilization results for the nonlinear wave equation can be found in [2]and the references therein. Some works have been done in the difficult critical case p = 5,we refer to [14] and [38].
The proofs of these articles use three main ingredients:
(i) the exponential decay of the linear equation, which is equivalent to the geometric controlcondition (GCC),
(ii) a more or less involved compactness argument,
(iii) a unique continuation result implying that u ≡ 0 is the unique solution of
u+ βu+ f(u) = 0
∂tu = 0 on [−T, T ]× ω .(1.8)
The results are mainly of the type: “geometric control condition” + “unique continuation”implies “exponential decay”. This type of implication is even stated explicitly in somerelated works for the nonlinear Schrodinger equation [15] and [37].
In the subcritical case p < 5, the less understood point is the unique continuationproperty (iii). In the previous works as [17], the authors use unique continuation resultsbased on Carleman estimates. The resulting geometric assumptions are not very naturaland are stronger than (GCC). Indeed, the unique continuation was often proved with someCarleman estimates that required some strong geometric conditions. For instance for aflat metric, the usual geometric assumption that appear are often of “multiplier type” thatis ω is a neighbourhood of x ∈ ∂Ω |(x− x0) · n(x) > 0 which are known to be strongerthan the geometric control condition (see [41] for a discussion about the links betweenthese assumptions). Moreover, on curved spaces, this type of condition often needs to bechecked by hand in each situation, which is mostly impossible.
Our main improvement in this paper is the proof of unique continuation in infinite timeunder the geometric control condition only. We show that, if the nonlinearity f is analytic(or generic), then one can use the result of Robbiano and Zuily [47] to obtain a uniquecontinuation property (iii) for infinite time T = +∞ with the geometric control condition(GCC) only.
The central argument of the proof of our main result, Theorem 1.1, is the uniquecontinuation property of [47] (see Section 3). This result applies for solutions u of (1.8)being smooth in space and analytic in time. If f is analytic, then the solutions of (1.1) are ofcourse not necessarily analytic in time since the damped wave equations are not smoothing
5
in finite time. However, the damped wave equations admit an asymptotic smoothing effect,i.e. are smoothing in infinite time. Hale and Raugel have shown in [25] that, for compacttrajectories, this asymptotic smoothing effect also concerns the analyticity (see Section 5).In other words, combining [47] and [25] shows that the unique solution of (1.8) is u ≡ 0 iff is analytic and if T = +∞. This combination has already been used by dynamicians in[26] and [32] for p < 3.
One of the main interests of this paper is the use of arguments coming from boththe dynamical study and the control theory of the damped wave equations. The readerfamiliar with the control theory could find interesting the use of the asymptotic smoothingeffect to get unique continuation property with smooth solutions. The one familiar with thedynamical study of PDEs could be interested in the use of Strichartz estimates to deal withthe case p ∈ [3, 5). The main part of the proof of Theorem 1.1 is written with argumentscoming from the dynamical study of PDEs. They are simpler than the corresponding onesof control theory, but far less accurate since they do not give any estimation for the timeof observability. Anyway, such accuracy is not important here since we use the uniquecontinuation property for (1.8) with T = +∞. We briefly recall in Section 8 how thesepropagation of compactness and regularity properties could have been proved with somearguments more usual in the control theory.
Moreover, we give two applications of our results in both contexts of control theory anddynamical systems. First, as it is usual in control theory, some results of stabilization canbe coupled with local control theorems to provide global controllability in large time.
Theorem 1.3. Assume that f satisfies the conditions of Theorem 1.1 or belongs to thegeneric set G defined by Theorem 1.2. Let R0 > 0 and ω satisfying the geometric controlcondition. Then, there exists T > 0 such that for any (u0, u1) and (u0, u1) in H1
0 (Ω)×L2(Ω),with
‖(u0, u1)‖H1×L2 ≤ R0; ‖(u0, u1)‖H1×L2 ≤ R0
there exists g ∈ L∞([0, T ], L2(Ω)) supported in [0, T ]×ω such that the unique strong solutionof
u+ βu+ f(u) = g on [0, T ]× Ω
(u(0), ∂tu(0)) = (u0, u1).
satisfies (u(T ), ∂tu(T )) = (u0, u1).
The second application of our results concerns the existence of a compact global attrac-tor. A compact global attractor is a compact set, which is invariant by the flow of the PDEand which attracts the bounded sets. The existence of such an attractor is an importantdynamical property because it roughly says that the dynamics of the PDE may be reduced
6
to dynamics on a compact set, which is often finite-dimensional. See for example [24] and[45] for a review on this concept. Theorems 1.1 and 1.2 show that 0 is a global attractorfor the damped wave equation (1.1). Of course, it is possible to obtain a more complexattractor by considering an equation of the type
∂2ttu+ γ(x)∂tu = ∆u− βu− f(x, u) (x, t) ∈ Ω× R+ ,
u(x, t) = 0 (x, t) ∈ ∂Ω × R+
(u, ∂tu) = (u0, u1) ∈ H10 × L2
(1.9)
where f ∈ C∞(Ω × R,R) is real analytic with respect to u and satisfies the followingproperties. There exist C > 0, p ∈ [1, 5) and R > 0 such that for all (x, u) ∈ Ω× R,
|f(x, u)| ≤ C(1 + |u|)p , |f ′x(x, u)| ≤ C(1 + |u|)p , |f ′
u(x, u)| ≤ C(1 + |u|)p−1 (1.10)
(x 6∈ B(x0, R) or |u| ≥ R) =⇒ f(x, u)u ≥ 0 . (1.11)
where x0 denotes a fixed point of the manifold.
Theorem 1.4. Assume that f is as above. Then, the dynamical system generated by (1.9)in H1
0 (Ω)× L2(Ω) is gradient and admits a compact global attractor A.
Of course, we would get the same result for f in a generic set similar to the one ofTheorem 1.2.
We begin this paper by setting our main notations and recalling the basic properties ofEquation (1.1) in Section 2. We recall the unique continuation property of Robbiano andZuily in Section 3, whereas Sections 4 and 5 are concerned by the asymptotic compactnessand the asymptotic smoothing effect of the damped wave equation. The proofs of our mainresults, Theorem 1.1 and 1.2, are given in Sections 6 and 7 respectively. An alternativeproof, using more usual arguments from control theory, is sketched in Section 8. Finally,Theorems 1.3 and 1.4 are discussed in Section 9.
Acknowledgements: the second author was financed by the ERC grant GeCoMethodsduring part of the redaction of this article. Moreover, both authors benefited from thefruitful atmosphere of the conference Partial differential equations, optimal design andnumerics in Benasque. We also would like to thank Mathieu Leautaud for his remarksabout the optimality of Hypothesis (GCC) in the nonlinear context and Genevieve Raugelfor her help for removing a non-natural hypothesis of Theorem 1.4.
2 Notations and basic properties of the damped wave
equation
In this paper, we use the following notations:
U = (u, ut) , F = (0, f) , A =
(0 Id
∆− β −γ
).
7
In this setting, (1.1) becomes
∂tU(t) = AU(t) + F (U) .
We set X = H10 (Ω)×L2(Ω) and for s ∈ [0, 1], Xs denotes the space D((−∆+ β)(s+1)/2)×
D((−∆ + β)s/2) = (H1+s(Ω) ∩ H10 (Ω)) × Hs
0(Ω). Notice that X0 = X and X1 = D(A)(even if γ is only in L∞).
We recall that E denotes the energy defined by (1.4). We also emphasize that (1.2)and the invertibility of ∆− β implies that a set is bounded in X if and only if its energyE is bounded. Moreover, for all E0 ≥ 0, there exists C > 0 such that
∀(u, v) ∈ X, E(u, v) ≤ E0 =⇒ 1
C‖(u, v)‖2X ≤ E(u, v) ≤ C ‖(u, v)‖2X (2.1)
To simplify some statements in the proofs, we assume without loss of generality that3 < p < 5. It will avoid some meaningless statements with negative Lebesgue exponentssince p = 3 is the exponent where Strichartz estimates are no more necessary and can bereplaced by Sobolev embeddings.
We recall that Ω is endowed with a metric g. We denote by d the distance on Ω definedby
d(x, y) = inf l(c) |c ∈ C∞([0, 1],Ω) with c(0) = x and c(1) = ywhere l(c) is the length of the path c according to the metric g. A ball B(x,R) in Ω isnaturally defined by
B(x,R) = y ∈ Ω, d(x, y) < R .
For instance, if Ω = R3 \ BR3(0, 1), the distance between (0, 0, 1) and (0, 0,−1) is π (and
not 2) and the ball B((0, 0, 1), π) has nothing to do with the classical ball BR3((0, 0, 1), π)of R3.
2.1 Cauchy problem
The global existence and uniqueness of solutions of the subcritical wave equation (1.1) withγ ≡ 0 has been studied by Ginibre and Velo in [21] and [22]. Their method also appliesfor γ 6= 0 since this term is linear and well defined in the energy space X . Moreover, theirargument to prove uniqueness also yields the continuity of the solutions with respect tothe initial data.
The central argument is the use of Strichartz estimates.
Theorem 2.1 (Strichartz estimates).Let T > 0 and (q, r) satisfying
1
q+
3
r=
1
2, q ∈ [7/2,+∞]. (2.2)
8
There exists C = C(T, q) > 0 such that for every G ∈ L1([0, T ], L2(Ω)) and every (u0, u1) ∈X, the solution u of
u+ γ(x)∂tu = G(t)
(u, ∂tu)(0) = (u0, u1)
satisfies the estimate
‖u‖Lq([0,T ],Lr(Ω)) ≤ C(‖u0‖H1(Ω) + ‖u1‖L2(Ω) + ‖G‖L1([0,T ],L2(Ω))
).
The result was stated in the Euclidean space R3 by Strichartz [48] and Ginibre and
Velo with q ∈ (2,+∞]. Kapitanski extended the result to variable coefficients in [33]. Ona bounded domain, the first estimates were proved by Burq, Lebeau and Planchon [12]for q ∈ [5,+∞] and extended to a larger range by Blair, Smith and Sogge in [8]. Notethat, thanks to the counterexamples of Ivanovici [31], we know that we cannot expect someStrichartz estimates in the full range of exponents in the presence of boundaries.
From these results, we deduce the estimates for the damped wave equation by ab-sorption for T small enough. We can iterate the operation in a uniform number ofsteps. Actually, for the purpose of the semilinear wave equation, it is sufficient to con-
sider the Strichartz estimate L2p
p−3 ([0, T ], L2p(Ω)) which gives up ∈ L2
p−3 ([0, T ], L2(Ω)) ⊂L1([0, T ], L2(Ω)) because 1 < 2
p−3< +∞.
Theorem 2.2 (Cauchy problem).Let f satisfies (1.2). Then, for any (u0, u1) ∈ X = H1
0 (Ω) × L2(Ω) there exists a uniquesolution u(t) of the subcritical damped wave equation (1.1). Moreover, this solution isdefined for all t ∈ R and its energy E(u(t)) is non-increasing in time.
For any E0 ≥ 0, T ≥ 0 and (q, r) satisfying (2.2), there exists a constant C such that,if u is a solution of (1.1) with E(u(0)) ≤ E0, then
‖u‖Lq([0,T ],Lr(Ω)) ≤ C(‖u0‖H1(Ω) + ‖u1‖L2(Ω)
).
In addition, for any E0 ≥ 0 and T ≥ 0, there exists a constant C such that, if u and uare two solutions of (1.1) with E(u(0)) ≤ E0 and E(u(0)) ≤ E0, then
supt∈[−T,T ]
‖(u, ∂tu)(t)− (u, ∂tu)(t)‖X ≤ C‖(u, ∂tu)(0)− (u, ∂tu)(0)‖X .
Proof: The existence and uniqueness for small times is a consequence of Strichartz esti-mates and of the subcriticality of the nonlinearity, see [22]. The solution can be globalizedbackward and forward in time thanks to the energy estimates (1.5) for smooth solutions.Indeed,
E(t) ≤ E(s) + C
∫ s
t
E(τ) dτ
9
and thus, Gronwall inequality for t ≤ s and the decay of energy for t ≥ s show that theenergy does not blow up in finite time. This allows to extend the solution for all timessince the energy controls the norm of the space X by (2.1).
For the uniform continuity estimate, we notice that w = u− u is solution of
w + βw + γ(x)∂tw = −wg(u, u)
(w, ∂tw)(0) = (u, ∂tu)(0)− (u, ∂tu)(0)
where g(s, s) =∫ 1
0f ′(s+ τ(s− s)) dτ fulfills |g(s, s)| ≤ C(1 + |s|p−1 + |s|p−1). Let q = 2p
p−3,
Strichartz and Holder estimates give
‖(w, ∂tw)(t)‖L∞([0,T ],X)∩Lq([0,T ],L2p) ≤ C‖(w, ∂tw)(0)‖X + C ‖wg(u, u)‖L1([0,T ],L2)
≤ C‖(w, ∂tw)(0)‖X + CT ‖w‖L∞([0,T ],L2)
+T θ ‖w‖Lq([0,T ],L2p)
(‖u‖p−1
Lq([0,T ],L2p) + ‖u‖p−1Lq([0,T ],L2p)
)
with θ = 5−p2
> 0. We get the expected result for T small enough by absorption since wealready know a uniform bound (depending on E0) for the Strichartz norms of u and u.Then, we iterate the operation to get the result for large T .
2.2 Exponential decay of the linear semigroup
In this paper, we will strongly use the exponential decay for the linear semigroup in thecase where γ may vanish but satisfies the geometric assumptions of this paper. In thiscase, (1.3) enables to control the decay of energy outside a large ball and the geometriccontrol condition (GCC) enables to control the energy trapped in this ball.
Proposition 2.3. Assume that γ ∈ L∞(Ω) satisfies (1.3) and (GCC). There exist twopositive constants C and λ such that
∀s ∈ [0, 1] , ∀t ≥ 0 , |||eAt|||L(Xs) ≤ Ce−λt .
The exponential decay of the damped wave equation under the geometric control con-dition is well known since the works of Rauch and Taylor on a compact manifold [44] andBardos, Lebeau and Rauch [7, 6] on a bounded domain. Yet, we did not find any referencefor unbounded domain ([3] and [35] concern unbounded domains but local energy only).It is noteworthy that the decay of the linear semigroup in unbounded domains seems notto have been extensively studied for the moment.
We give a proof of Proposition 2.3 using microlocal defect measure as done in Lebeau[39] or Burq [10] (see also [11] for the proof of the necessity). The only difference with
10
respect to these results is that the manifold that we consider may be unbounded. Sincemicrolocal defect measure only reflects the local propagation, we thus have to use theproperty of equipartition of the energy to deal with the energy at infinity and to show apropagation of compactness (see [17] for the flat case).
Lemma 2.4. Let T > L where L is given by (GCC). Assume that (Un,0) ⊂ X is a boundedsequence, which weakly converges to 0 and assume that Un(t) = (un(t), ∂tun(t)) = eAtUn,0
satisfies∫ T
0
∫
Ω
γ(x)|∂tun|2 → 0 . (2.3)
Then, (Un,0) converges to 0 strongly in X.
Proof: Let µ be a microlocal defect measure associated to (un) (see [18], [50] or [9] forthe definition). Note that (2.3) implies that µ can also be associated to the solution ofthe wave equation without damping, so the weak regularity of γ is not problematic for thepropagation and we get that µ is concentrated on τ 2 − |ξ|2x = 0 where (τ, ξ) are the dualvariables of (t, x). Moreover, (2.3) implies that γτ 2µ = 0 and so µ ≡ 0 on S∗(]0, T [×ω).Then, using the propagation of the measure along the generalized bicharacteristic flow ofMelrose-Sjostrand and the geometric control condition satisfied by ω, we obtain µ ≡ 0 ev-erywhere. We do not give more details about propagation of microlocal defect measure andrefer to the Appendix [39] or Section 3 of [9] (see also [20] for some close propagation resultsin a different context). Since µ ≡ 0, we know that Un → 0 on H1 × L2(]0, T [×B(x0, R))for every R > 0.
To finish the proof, we need the classical equipartition of the energy to get the conver-gence to 0 in the whole manifold Ω. Since γ is uniformly positive outside a ball B(x0, R),(2.3) and the previous arguments imply that ∂tun → 0 in L2([0, T ]×Ω). Let ϕ ∈ C∞
0 (]0, T [)with ϕ ≥ 0 and ϕ(t) = 1 for t ∈ [ε, T − ε]. We multiply the equation by ϕ(t)un and weobtain
0 = −∫∫
[0,T ]×Ω
ϕ(t)|∂tun|2 −∫∫
[0,T ]×Ω
ϕ′(t)∂tunun +
∫∫
[0,T ]×Ω
ϕ(t)|∇un|2
+
∫∫
[0,T ]×Ω
ϕ(t)β|un|2 +∫∫
[0,T ]×Ω
ϕ(t)γ(x)∂tunun.
The L2−norm of un(t) is bounded, while ∂tun → 0 in L2([0, T ] × Ω), so the first, secondand fifth terms converge to zero. Then, the above equation yields
∫∫
[0,T ]×Ω
ϕ(t)(β|un|2 + |∇un|2
)−→ 0.
Finally, notice that the energy identity ‖Un,0‖2X = ‖Un(t)‖2X +∫ T
0
∫Ωγ(x)|∂tun|2 shows that
∫∫
[0,T ]×Ω
ϕ(t)(β|un|2 + |∇un|2
)∼ ‖Un,0‖2X
∫ T
0
ϕ(t)
11
and thus that ‖Un,0‖X goes to zero.
Proof of Proposition 2.3: Once Lemma 2.4 is established, the proof follows the argu-ments of the classical case, where Ω is bounded. We briefly recall them.
We first treat the case s = 0. As in Proposition 2.5, the exponential decay of the energyis equivalent to the observability estimate, that is the existence of C > 0 and T > 0 suchthat, for any trajectory U(t) = eAtU0 in X ,
∫ T
0
∫
Ω
γ(x)|∂tu|2 ≥ C ‖U(0)‖2X .
We argue by contradiction: assume that (2.4) does not hold for any positive T and C.Then, there exists a sequence of initial data Un(0) with ‖Un(0)‖X = 1 and such that
∫ n
0
∫
Ω
γ(x)|∂tun(t, x)|2dtdx −−−−−−−→n−→+∞
0 ,
where (un, ∂tun)(t) = Un(t) = eAtUn(0). Let Un = Un(n/2 + ·). We have
∫ n/2
−n/2
∫
Ω
γ(x)|∂tun(t, x)|2dtdx −−−−→n→∞
0 ,
and, for any t ∈ [−n/2, n/2],
‖Un(t)‖2X = ‖Un(−n/2)‖2X −∫ t
−n/2
∫
Ω
γ(x)|∂tun(s, x)|2dsdx −−−−→n→∞
1 .
Therefore, we can assume that Un(0) converges to U∞(0) ∈ X , weakly in X . Moreover,for any T > 0, Un(t) and ∂tUn(t) are bounded in L∞([−T, T ], X) and L∞([−T, T ], L2(Ω)×H−1(Ω)) respectively. Thus, using Ascoli’s Theorem, we may also assume that Un(t)strongly converges to U∞(t) in L∞([−T, T ], L2(K)×H−1(K)) where K is any compact ofΩ. Hence, (u∞, ∂tu∞)(t) = U∞(t) = eAtU∞(0) is a solution of
u∞ + βu∞ = 0 on R× Ω
∂tu∞ = 0 on R× ω.(2.4)
in L2 ×H−1. Since U∞(0) ∈ X belongs to X , we deduce that, in fact, U∞(t) solves (2.4)in X .
To finish the proof of Proposition 2.3, we have to show that U∞ ≡ 0. Indeed, applyingLemma 2.4, we would get that Un converges strongly to 0, which contradicts the hypothesis‖Un(0)‖X = 1. Note that U∞ ≡ 0 is a direct consequence of a unique continuation propertyas Corollary 3.2. However, Corollary 3.2 requires Ω to be smooth, whereas Proposition
12
2.3 could be more general. Therefore, we recall another classical argument to show thatU∞ ≡ 0.
Denote N the set of function U∞(0) ∈ X satisfying (2.4), which is obviously a linearsubspace of X . We will prove that N = 0. Since γ(x)|∂tu∞|2 ≡ 0 for functions u∞ in Nand since N is a closed subspace, Lemma 2.4 shows that any weakly convergent subsequenceof N is in fact strongly convergent. By Riesz Theorem, N is therefore finite dimensional.For any t ∈ R, etA applies N into itself and thus A|N is a bounded linear operator. Assumethat N 6= 0, then A|N admits an eigenvalue λ with eigenvector Y = (y0, y1) ∈ N . Thismeans that y1 = λy0 and that (∆−β)y0 = λ2y0. Moreover, we know that y1 = 0 on ω andso, if λ 6= 0, that y0 = 0 on ω. This implies y0 ≡ 0 by the unique continuation propertyof elliptic operators. Finally, if λ = 0, we have (∆ − β)y0 = 0 and y0 = 0, because, byassumption, ∆− β is a negative definite operator.
So we have proved N = 0 and therefore U∞ = 0, that is Un(0) converges to 0 weaklyin X . We can then apply Lemma 2.4 on any interval [−n/2,−n/2+T ] where L is the timein the geometric control condition (GCC) and obtain a contradiction to ‖Un(0)‖X = 1.
Let us now consider the cases s ∈ (0, 1]. The basic semigroup properties (see [43])shows that, if U ∈ X1 = D(A), then eAtU belongs to D(A) and
‖eAtU‖X1 = ‖AeAtU‖X + ‖eAtU‖X = ‖eAtAU‖X + ‖eAtU‖X≤ Ce−λt (‖AU‖X + ‖U‖X) = Ce−λt‖U‖D(A) .
This shows Proposition 2.3 for s = 1. Notice that we do not have to require any regularityfor γ to obtain this result. Then, Proposition 2.3 for s ∈ (0, 1) follows by interpolatingbetween the cases s = 0 and s = 1 (see [49]).
2.3 First nonlinear exponential decay properties
Theorem 2.2 shows that the energy E is non-increasing along the solutions of (1.1). Thepurpose of this paper is to obtain the exponential decay of this energy in the sense ofproperty (ED) stated above. We first recall the well-known criterion for exponential decay.
Proposition 2.5. The exponential decay property (ED) holds if and only if, there exist Tand C such that
E(u(0)) ≤ C(E(u(0))− E(u(T ))) = C
∫ ∫
[0,T ]×Ω
γ(x)|∂tu(x, t)|2dtdx (2.5)
for all solutions u of (1.1) with E(u(0)) ≤ E0.
Proof: If (ED) holds then obviously (2.5) holds for T large enough since E(u(0)) −E(u(T )) ≥ (1 −Ke−λT )E(u(0)). Conversely, if (2.5) holds, using E(u(T )) ≤ E(u(0)), we
13
get E(u(T )) ≤ C/(C+1)E(u(0)) and thus E(u(kT )) ≤ (C/(C+1))kE(u(0)). Using againthe decay of the energy to fill the gaps t ∈ (kT, (k + 1)T ), this shows that (ED) holds.
First, we prove exponential decay in the case of positive damping, which will be helpfulto study what happens outside a large ball since (1.3) is assumed in the whole paper.Note that the fact that −∆+ β is positive is necessary to avoid for instance the constantundamped solutions.
Proposition 2.6. Assume that ω = Ω, that is that γ(x) ≥ α > 0 everywhere. Then (ED)holds.
Proof: We recall here the classical proof. We introduce a modified energy
E(u) =
∫
Ω
1
2(|∂tu|2 + |∇u|2 + β|u|2) + V (u) + εu∂tu
with ε > 0. Since∫Ω|∇u|2 + β|u|2 controls ‖u‖2L2, E is equivalent to E for ε small enough
and it is sufficient to obtain the exponential decay of the auxiliary energy E. Usingγ ≥ α > 0 and uf(u) ≥ 0, a direct computation shows for ε small enough that
E(u(T ))− E(u(0)) =
∫ T
0
∫
Ω
−γ(x)|∂tu|2 + ε|∂tu|2 + εγ(x)u∂tu− ε(|∇u|2 + β|u|2)− εuf(u)
≤ −C
∫ T
0
‖(u, ∂tu)‖2H1×L2
≤ −C
∫ T
0
E(t)dt
≤ −CTE(T ) ,
where C is some positive constant, not necessarily the same from line to line. Thus,E(u(0)) − E(u(T )) ≥ CTE(u(T )) with CT > 0 and therefore E(u(0)) ≥ µE(u(T )) withµ > 1. As in the proof of Proposition 2.3, this last property implies the exponential decayof E and thus the one of E.
3 A unique continuation result for equations with par-
tially holomorphic coefficients
Comparatively to previous articles on the stabilization of the damped wave equations as[17], one of the main novelties of this paper is the use of a unique continuation theorem
14
requiring partially analyticity of the coefficients, but very weak geometrical assumptionsas shown in Corollary 3.2. We use here the result of Robbiano and Zuily in [47]. Thisresult has also been proved independently by Hormander in [30] and has been generalisedby Tataru in [52]. Note that the idea of using partial analyticity for unique continuationwas introduced by Tataru [51] but it requires some global analyticity assumptions that arenot fulfilled in our case. All these results use very accurate microlocal analysis and holdin a much more general framework than the one of the wave equation. However, for sakeof simplicity, we restrict the statement to this case.
Theorem 3.1. Robbiano-Zuily, Hormander (1998)Let d ≥ 1, (x0, t0) ∈ R
d×R and let U be a neighbourhood of (x0, t0). Let (Ai,j(x, t))i,j=1,...,d,b(x, t), (ci(x, t))i=1,...,d) and d(x, t) be bounded coefficients in C∞(U ,R). Let v be a strongsolution of
∂2ttv = div(A(x, t)∇v) + b(x, t)∂tv + c(x, t).∇v + d(x, t)v (x, t) ∈ U ⊂ R
d × R . (3.1)
Let ϕ ∈ C2(U ,R) such that ϕ(x0, t0) = 0 and (∇ϕ, ∂tϕ)(x, t) 6= 0 for all (x, t) ∈ U . Assumethat:(i) the coefficients A, b, c and d are analytic in time,(ii) A(x0, t0) is a symmetric positive definite matrix,(iii) the hypersurface (x, t) ∈ U , ϕ(x, t) = 0 is not characteristic at (x0, t0) that is thatwe have |∂tϕ(x0, t0)|2 6= 〈∇ϕ(x0, t0)|A(x0, t0)∇ϕ(x0, t0)〉(iv) v ≡ 0 in (x, t) ∈ U , ϕ(x, t) ≤ 0.Then, v ≡ 0 in a neighbourhood of (x0, t0).
Proof: We only have to show that Theorem 3.1 is a direct translation of Theorem A of[47] in the framework of the wave equation. To use the notations of [47], we set xa to bethe time variable and xb the space variable and we set (x0, t0) = x0 = (x0
b , x0a). Equation
(3.1) corresponds to the differential operator
P = ξ2a − tξbA(xb, xa)ξb − b(xb, xa)ξa − c(xb, xa)ξb − d(xb, xa)
with principal symbol p2 = ξ2a − tξbA(xb, xa)ξb.All the statement of Theorem 3.1 is an obvious translation of Theorem A of [47], except
maybe for the fact that Hypothesis (iii) implies the hypothesis of pseudo-convexity of [47].We compute p2, ϕ = 2ξaϕ
′a−2tξbA(xa, xb)ϕ
′b. Let us set ζ = (x0
a, x0b , iϕ
′a(x
0), ξb+iϕ′b(x
0)),then p2, ϕ (ζ) = 0 if and only if
i(ϕ′a(x
0))2 − itϕ′b(x
0))A(x0)ϕ′b(x
0)−t ξbA(x0)ϕ′
b(x0) = 0 .
This is possible only if (ϕ′a(x
0))2 = tϕ′b(x
0)A(x0)ϕ′b(x
0), that is if the hypersurface ϕ = 0 ischaracteristic at (x0, t0). Thus, if this hypersurface is not characteristic, then the pseudo-convexity hypothesis of Theorem A of [47] holds.
15
The previous theorem allows to prove some unique continuation result with some opti-mal time and geometric assumption. This allows to prove unique continuation where thegeometric condition is only, roughly speaking, that we do not contradict the finite speedof propagation.
Corollary 3.2. Let T > 0 (or T = +∞) and let b, (ci)i=1,2,3 and d be smooth coefficientsin C∞(Ω × [0, T ],R). Assume moreover that b, c and d are analytic in time and that v isa strong solution of
∂2ttv = ∆v + b(x, t)∂tv + c(x, t).∇v + d(x, t)v (x, t) ∈ Ω× (−T, T ) . (3.2)
Let O be a non-empty open subset of Ω and assume that v(x, t) = 0 in O× (−T, T ). Thenv(x, 0) ≡ 0 in OT = x0 ∈ Ω , d(x0,O) < T.As consequences:a) if T = +∞, then v ≡ 0 everywhere,b) if v ≡ 0 in O × (−T, T ) and OT = Ω, then v ≡ 0 everywhere.
Proof: Since Ω is assumed to be connected, both consequences are obvious from the firststatement.
Let x0 be given such that d(x0,O) < T . There is a point x∗ ∈ O linked to x0 bya smooth curve of length l < T , which stays away from the boundary. We introduce asequence of balls B(x0, r), . . . , B(xK , r) with r ∈ (0, T/K), xk−1 ∈ B(xk, r) and xK = x∗,such that B(xk, r) stays away from the boundary and is small enough such that it isdiffeomorphic to an open set of R3 via the exponential map. Note that such a sequence ofballs exists because the smooth curve linking x0 to xK is compact and of length smallerthan T . We also notice that it is sufficient to prove Corollary 3.2 in each ball B(xk, r).Indeed, this would enable us to apply Corollary 3.2 in B(xK , r)× (−T, T ) to obtain that vvanishes in a neighbourhood of xK−1 for t ∈ (−T +r, T−r) and then to apply it recursivelyin B(xK−1, r)× (−T + r, T − r), . . . , B(x1, r)× (−T + (K − 1)r, T − (K − 1)r) to obtainthat v(x0, 0) = 0.
From now on, we assume that x0 ∈ B(x∗, r) and that v vanishes in a neighbourhoodO of x∗ for t ∈ (−r, r). Since d(x0, x∗) < r, we can introduce a non-negative functionh ∈ C∞([−r, r],R) such that h(0) > d(x0, x∗), h(±r) = 0 and |h′(t)| < 1 for all t ∈ [−r, r].We set U = B(x∗, r)× (−r, r) and for any λ ∈ [0, 1], we define
ϕλ(x, t) = d(x, x∗)2 − λh(t)2 .
Since r is assumed to be smaller than the radius of injectivity of the exponential map, ϕλ
is a smooth well-defined function. We prove Corollary 3.2 by contradiction. Assume thatv(x0, 0) 6= 0. We denote by Vλ the volume (x, t) ∈ U , ϕλ(x, t) ≤ 0. We notice thatVλ1
⊂ Vλ2if λ1 < λ2, that for small λ, Vλ is included in O × (−r, r) where v vanishes, and
that V1 contains (x0, 0) where v does not vanish. Thus
λ0 = supλ ∈ [0, 1] , ∀(x, t) ∈ Vλ, v(x, t) = 0
16
is well defined and belongs to (0, 1). For t close to −r or r, h(t) is small and the sectionx, (x, t) ∈ Vλ0
of Vλ0is contained in O where v vanishes. Therefore, by compactness,
the hypersurface Sλ0= ∂Vλ0
must touch the support of v at some point (x1, t1) ∈ U (seeFigure 1).
O
supp(u)
t
x
(x1, t1)Sλ
Sλ0
(x0, 0)
Figure 1: The proof of Corollary 3.2
In local coordinates, ∆ can be written div(A(x)∇.)+c(x)·∇. Moreover, 〈∇ϕλ|A∇ϕλ〉 =|∇gd(., x∗)|2g = 1 where the index g means that the gradient and norm are taken accordingto the metric. Therefore, the hypersurface Sλ0
is non-characteristic at (x1, t1) in the senseof Hypothesis (iii) of Theorem 3.1 since |∂tϕλ(x, t)| = |λh′(t1)| < 1. Thus, we can applyTheorem 3.1 with ϕ = ϕλ0
at the point (x1, t1), mapping everything in the 3d-euclideanframe via the exponential chart. We get that v must vanish in a neighbourhood of (x1, t1).This is obviously a contradiction since (x1, t1) has been taken in the support of v.
4 Asymptotic compactness
As soon as t is positive, a solution u(t) of a parabolic PDE becomes smooth and stays ina compact set. The smoothing effect in finite time of course fails for the damped waveequations. However, these PDEs admit in some sense a smoothing effect in infinite time.This effect is called asymptotic compactness if one is interested in extracting asymptoticsubsequences as in Proposition 4.3, or asymptotic smoothness if one uses the regularity ofglobally bounded solutions as in Proposition 4.4. For the reader interested in these notions,
17
we refer for example to [24]. The proof of this asymptotic smoothing effect is based on thevariation of constant formula U(t) = eAtU0 +
∫ t
0eA(t−s)F (U(s))ds and two properties:
- the exponential decay of the linear group (Proposition 2.3), which implies that the linearpart eAtU0 asymptotically disappears,- the regularity of the nonlinearity F implying the compactness of the nonlinear term∫ t
0eA(t−s)F (U(s))ds (Corollary 4.2 below). Note that the subcriticality of f is the key
point of this property and that our arguments cannot be extended as they stand to thecritical case p = 5.
The purpose of this section is to prove some compactness and regularity results aboutundamped solutions as (1.8). Note that these results could also have been obtained witha more “control theoretic” proof (see section 8 for a sketch of the alternative proof) basedon propagation results or observability estimates. Here, we have chosen to give a differentone using asymptotic regularization, more usual in dynamical system. The spirit of theproof remains quite similar: prove that the nonlinearity is more regular than it seems apriori and use some properties of the damped linear equation.
4.1 Regularity of the nonlinearity
Since f is subcritical, it is shown in [17] that the nonlinear term of 1.1 yields a gain ofsmoothness.
Theorem 4.1. Dehman, Lebeau and Zuazua (2003)Let χ ∈ C∞
0 (R3,R), R > 0 and T > 0. Let s ∈ [0, 1) and let ε = min(1− s, (5− p)/2, (17−3p)/14) > 0 with p and f as in (1.2). There exist (q, r) satisfying (2.2) and C > 0 such thatthe following property holds. If v ∈ L∞([0, T ], H1+s(R3)) is a function with finite Strichartznorms ‖v‖Lq([0,T ],Lr(R3)) ≤ R, then χ(x)f(v) ∈ L1([0, T ], Hs+ε(R3)) and moreover
‖χ(x)f(v)‖L1([0,T ],Hs+ε(R3)) ≤ C‖v‖L∞([0,T ],H1+s(R3)) .
The constant C depends only on χ, s, T , (q, r), R and the constant in Estimate (1.2).
Theorem 4.1 is a copy of Theorem 8 of [17], except for two points.First, we would like to apply the result to a solution v of the damped wave equation on
a manifold possibly with boundaries, where not all Strichartz exponents are available. Thisleads to the constraint q ≥ 7/2 for the Strichartz exponents (q, r) of (2.2) (see Theorem2.2). In the proof of Theorem 8 of [17], the useful Strichartz estimate corresponds to
r = 3(p−1)1−ε
and q = 2(p−1)p−3+2ε
and it is required that q ≥ p− 1, which yields ε ≤ (5− p)/2. In
this paper, we require also that q ≥ 7/2 which yields in addition ε ≤ (17− 3p)/14. Noticethat p < 5 and thus both bounds are positive.
The second difference is that, in [17], the function f is assumed to be of class C3 andto satisfy
|f ′′(u)| ≤ C(1 + |u|)p−2 and |f (3)(u)| ≤ C(1 + |u|)p−3 (4.1)
18
in addition of (1.2). Since Theorem 4.1 concerns the L1(Hs′)−norm of χ(x)f(v) for s′ =s+ε ≤ 1, we can omit Assumption (4.1). Actually, we make the assumption ε ≤ 1−s whichis not present in [17] and a careful study of their proof shows that (1.2) is not necessaryunder that assumption.
Indeed, let f(u) = th3(u)|u|p. The function f is of class C3 and satisfies (1.2) and (4.1).Hence, Theorem 8 of [17] can be applied to f and we can bound the L1(Hs′)−norm of fas in Theorem 4.1. On the other hand, we notice that |f(u)| ∼
±∞|u|p, f ′(u) ∼
±∞p|u|p−1
and f ′(u) ≥ 0. Therefore, since f satisfies (1.2), there exists C > 0 such that |f(u)| ≤C(1 + |f(u)|) and |f ′(u)| ≤ C(1 + f ′(u)). Thus, if say v > u for fixing the notations,
|f(v)− f(u)| ≤ (v − u)
∫ 1
0
|f ′(u+ τ(v − u))|dτ
≤ C(v − u) + C(v − u)
∫ 1
0
f ′(u+ τ(v − u))dτ = C(v − u) + C(f(v)− f(u))
≤ C|v − u|+ C|f(v)− f(u)| .
For 0 < s < 1, using the above inequalities and the definition of the Hs′−norm as
‖χf(u)‖2Hs′ = ‖χf(u)‖2L2 +
∫∫
R6
|χ(x)f(u(x))− χ(y)f(u(y))|2|x− y|2s′ dxdy ,
we obtain‖χf(u)‖L1(Hs′ ) ≤ C‖u‖L∞(H1) + C‖χf(u)‖L1(Hs′ )
where χ is another cut-off function with larger support. Hence, for 0 < s < 1, theconclusion of Theorem 4.1 holds not only for f but also for f . If s′ = 1, we just apply thechain rule and the proof is easier.
Note that the above arguments show that the constant C depends on f through Esti-mate 1.2 only. Notice in addition that, since f is only C1, we cannot expect χf(v) to bemore regular than H1 and that is why we also assume ε ≤ 1− s.
In this paper, we use a generalisation of Theorem 8 of [17] to non-compact manifoldswith boundaries.
Corollary 4.2. Let R > 0 and T > 0. Let s ∈ [0, 1) and let ε = min(1−s, (5−p)/2, (17−3p)/14) > 0 with p as in (1.2). There exist (q, r) satisfying (2.2) and C > 0 such thatthe following property holds. If v ∈ L∞([0, T ], H1+s(Ω) ∩ H1
0 (Ω)) is a function with finiteStrichartz norms ‖v‖Lq([0,T ],Lr(Ω)) ≤ R, then f(v) ∈ L1([0, T ], Hs+ε
0 (Ω)) and moreover
‖f(v)‖L1([0,T ],Hs+ε0
(Ω)) ≤ C‖v‖L∞([0,T ],Hs+1(Ω)∩H10(Ω)) .
The constant C depends only on Ω, (q, r), R and the constant in Estimate (1.2).
19
Proof: Since we assumed that Ω has a bounded geometry in the sense that Ω is compactor a compact perturbation of a manifold with periodic metric, Ω can be covered by a setof C∞−charts αi : Ui 7−→ αi(Ui) ⊂ R
3 such that αi(Ui) is equal either to B(0, 1) or toB+(0, 1) = x ∈ B(0, 1), x1 > 0 and such that, for any s ≥ 0 the norm of a functionu ∈ Hs(Ω) is equivalent to the norm
(∑
i∈N
‖u α−1i ‖2Hs(αi(Ui))
)1/2 .
Moreover, the Strichartz norm Lq([0, T ], Lr(αi(Ui)) of v α−1i is uniformly controlled from
above by the Strichartz norm Lq([0, T ], Lr(Ui)) of v which is bounded by R.Therefore, it is sufficient to prove that Corollary 4.2 holds for Ω being either B(0, 1) or
B+(0, 1). Say that Ω = B+(0, 1), the case Ω = B(0, 1) being simpler. To apply Theorem4.1, we extend v in a neighbourhood of B+(0, 1) as follows. For x ∈ B+(0, 2), we use theradial coordinates x = (r, σ) and we set
v(x) = v(r, σ) = 5v(1− r, σ)− 20v(1− r/2, σ) + 16v(1− r/4, σ) .
Then, for x = (x1, x2, x3) ∈ B−(0, 2), we set
v(x) = 5v(−x1, x2, x3)− 20v(−x1/2, x2, x3) + 16v(−x1/4, x2, x3) .
Notice that v is an extension of v in B(0, 2), which preserves the C2−regularity, andthat the Hs−norm for s ≤ 2 as well as the Strichartz norms of v are controlled by thecorresponding norms of v. Let χ ∈ C∞
0 (R3) be a cut-off function such that χ ≡ 1 inB+(0, 1) and χ ≡ 0 outside B(0, 2). Applying Theorem 4.1 to χ(x)f(χ(x)v) yields acontrol of ‖f(v)‖L1([0,T ],Hs+ε(B+(0,1))) by ‖v‖L∞([0,T ],Hs+1(Ω)). Finally, notice that f(0) = 0and thus, the Dirichlet boundary condition on v naturally implies the one on f(v).
4.2 Asymptotic compactness and regularization effect
As explained in the beginning of this section, using Duhamel formula U(t) = eAtU0 +∫ t
0eA(t−s)F (U(s))ds and Corollary 4.2, we obtain two propositions related to the asymptotic
smoothing effect of the damped wave equations.
Proposition 4.3. Let f ∈ C1(R) satisfying (1.2), let (un0 , u
n1) be a sequence of initial data
which is bounded in X = H10 (Ω)×L2(Ω) and let (un) be the corresponding solutions of the
damped wave equation (1.1). Let (tn) ∈ R be a sequence of times such that tn → +∞ whenn goes to +∞.
Then, there exist subsequences (uφ(n)) and (tφ(n)) and a global solution u∞ of (1.1) suchthat
∀T > 0 , (uφ(n), ∂tuφ(n))(tφ(n) + .) −→ (u∞, ∂tu∞)(.) in C0([−T, T ], X) .
20
Proof: We use the notations of Section 2. Due to the equivalence between the norm ofX and the energy given by (2.1) and the fact that the energy is decreasing in time, weknow that Un(t) is uniformly bounded in X with respect to n and t ≥ 0. So, up to takinga subsequence, it weakly converges to a limit U∞(0) which gives a global solution U∞.We notice that, due to the continuity of the Cauchy problem with respect to the initialdata stated in Theorem 2.2, it is sufficient to show that Uφ(n)(tφ(n)) → U∞(0) for somesubsequence φ(n). We have
Un(tn) = eAtnUn(0) +
∫ tn
0
esAF (Un(tn − s)) ds
= eAtnUn(0) +
⌊tn⌋−1∑
k=0
ekA∫ 1
0
esAF (Un(tn − k − s)) ds+
∫ tn
⌊tn⌋
esAF (Un(tn − s)) ds
= eAtnUn(0) +
⌊tn⌋−1∑
k=0
ekAIk,n + In (4.2)
Theorem 2.2 shows that the Strichartz norms ‖un(tn − k − .)‖Lq([0,1],Lr(Ω)) are uniformlybounded since the energy of Un is uniformly bounded. Therefore, Corollary 4.2 and Propo-sition 2.3 show that the terms In,k =
∫ 1
0esAF (Un(tn−k−s)) ds, as well as In, are bounded
by some constant M in H1+ε(Ω) × Hε(Ω) uniformly in n and k. Using Proposition 2.3again and summing up, we get that the last terms of (4.2) are bounded in H1+ε(Ω)×Hε(Ω)uniformly in n by
∥∥∥∥∥∥
⌊tn⌋−1∑
k=0
ekAIk,n + In
∥∥∥∥∥∥Xε
≤⌊tn⌋−1∑
k=0
Ce−λkM +M ≤ M
(1 +
C
1− e−λ
).
Moreover, Proposition 2.3 shows that eAtnUn(0) goes to zero in X when n goes to +∞.Therefore, by a diagonal extraction argument and Rellich Theorem, we can extract asubsequence Uφ(n)(tφ(n)) that converges to U∞(0) in H1
0 (B)× L2(B) for all bounded set Bof Ω.
To finish the proof of Proposition 4.3, we have to show that this convergence holds infact in X and not only locally. Let η > 0 be given. Let T > 0 and let Un be the solution of(1.1) with Un(0) = Un(tn−T ) and with γ being replaced by γ, where γ(x) ≡ γ(x) for largex and γ ≥ α > 0 everywhere. By Proposition 2.6, ‖Un(T )‖X ≤ η if T is chosen sufficientlylarge and if n is large enough so that tn − T > 0. Since the information propagates atfinite speed in the wave equation, Un(tn) ≡ Un(T ) outside a large enough bounded set andthus Uφ(n)(tφ(n)) has a X−norm smaller than η outside this bounded set. On the otherhand, we can assume that the norm of U∞(0) is also smaller than η outside the boundedset. Then, choosing n large enough, ‖Uφ(n)(tφ(n))− U∞(0)‖X becomes smaller than 3η.
21
The trajectories U∞ appearing in Proposition 4.3 are trajectories which are bounded inX for all times t ∈ R. The following result shows that these special trajectories are moreregular than the usual trajectories of the damped wave equation.
Proposition 4.4. Let f ∈ C1(R) satisfying (1.2) and let E0 ≥ 0. There exists a constantM such that if u is a solution of (1.1), which exists for all times t ∈ R and satisfiessupt∈R E(u(t)) ≤ E0, then t 7→ U(t) = (u(t), ∂tu(t)) is continuous from R into D(A) and
supt∈R
‖(u(t), ∂tu(t))‖D(A) ≤ M .
In addition, M depends only on E0 and the constants in (1.2).
Proof: We use a bootstrap argument. For any t ∈ R and n ∈ N,
U(t) = enAU(t− n) +n−1∑
k=0
ekA∫ 1
0
esAF (U(t− k − s)) ds .
Using Proposition 2.3, when n goes to +∞, we get
U(t) =+∞∑
k=0
ekA∫ 1
0
esAF (U(t− k − s)) ds . (4.3)
Moreover, arguing exactly as in the proof of Proposition 4.3, we show that Proposition 2.3and Corollary 4.2 imply that Equality (4.3) also holds in Xε. Hence, U(t) is uniformlybounded in Xε. Then, using again Proposition 2.3 and Corollary 4.2, (4.3) also holds inX2ε etc. Repeating the arguments and noting that, until the last step, ε only depends onp, we obtain that U(t) is uniformly bounded in X1 = D(A).
Since the constant C of Corollary 4.2 only depends on f through Estimate (1.2), thesame holds for the bound M here.
Proposition 4.5. The Sobolev embedding H2(Ω) → C0(Ω) holds and there exists a con-stant K such that
∀u ∈ H2(Ω) , supx∈Ω
|u(x)| ≤ K‖u‖H2 .
In particular, the solution u in the statement of Proposition 4.4 belongs to C0(Ω × R,R)and sup(x,t)∈Ω×R
|u(x, t)| ≤ KM .
Proof: Proposition 4.5 follows directly from the fact that Ω has a bounded geometry andfrom the classical Sobolev embedding H2 → C0 in the ball B(0, 1) of R3.
22
5 Smoothness and uniqueness of non-dissipative com-
plete solutions
In this section, we consider only a non-dissipative complete solution, that is a solution u∗
existing for all times t ∈ R for which the energy E is constant. In other words, u∗(t) solves
∂2ttu
∗ = ∆u∗ − βu∗ − f(u∗) (x, t) ∈ Ω× R ,u∗(x, t) = 0 (x, t) ∈ ∂Ω× R
∂tu∗(x, t) = 0 (x, t) ∈ supp(γ)× R
(5.1)
Since the energy E is not dissipated by u∗(t), we can write E(u∗) instead of E(u∗(t)). Yet,an interesting fact that will be used several times in the sequel is that such u∗ is, at thesame time, solution of both damped and undamped equations.
The purpose of this section is:- First show that u∗ is analytic in time and smooth in space. The central argument is touse a theorem of J.K. Hale and G. Raugel in [25].- Then use the unique continuation result of L. Robbiano and C. Zuily stated in Corollary3.2 to show that u∗ is necessarily an equilibrium point of (1.1).- Finally show that the assumption sf(s) ≥ 0 imply that u∗ ≡ 0.
We enhance that the first two steps are valid and very helpful in a more general frame-work than the one of our paper.
5.1 Smoothness and partial analyticity of u∗
First, we recall here the result of Section 2.2 of [25], adapting the statement to suit ournotations.
Theorem 5.1. Hale and Raugel (2003)Let Y be a Banach space. Let Pn ∈ L(Y ) be a sequence of continuous linear maps and letQn = Id− Pn. Let A : D(A) → Y be the generator of a continuous semigroup etA and letG ∈ C1(Y ). We assume that V is a complete mild solution in Y of
∂tV (t) = AV (t) +G(V (t)) ∀t ∈ R .
We further assume that
(i) V (t), t ∈ R is contained in a compact set K of Y .(ii) for any y ∈ Y , Pny converges to y when n goes +∞ and (Pn) and (Qn) are sequencesof L(Y ) bounded by K0.(iii) the operator A splits in A = A1 +B1 where B1 is bounded and A1 commutes with Pn.(iv) there exist M and λ > 0 such that ‖eAt‖L(Y ) ≤ Me−λt and ‖e(A1+QnB1)t‖L(QnY,Y ) ≤Me−λt for all t ≥ 0.
23
(v) G is analytic in the ball BY (0, r), where r is such that r ≥ 4K0 supt∈R ‖V (t)‖Y . Moreprecisely, there exists ρ > 0 such that G can be extended to an holomorphic function ofBY (0, r) + iBY (0, ρ).(vi) DG(V (t))V2 |t ∈ R, ‖V2‖Y ≤ 1 is a relatively compact set of Y .
Then, the solution V (t) is analytic from t ∈ R into Y .
More precisely, Theorem 5.1 is Theorem 2.20 (which relates to Theorem 2.12) of [25]applied with Hypothesis (H3mod) and (H5).
Proposition 4.4 shows that u∗ is continuous in both space and time variables. We applyTheorem 5.1 to show that, because f is analytic, u∗ is also analytic with respect to thetime.
Proposition 5.2. Let f ∈ C1(R) satisfying (1.2) and let E0 ≥ 0. Let K and M be theconstants given by Propositions 4.4 and 4.5. Assume that f is analytic in [−4KM, 4KM ].Then for any non-dissipative complete solution u∗(t) solving (5.1) and satisfying E(u∗) ≤E0, t 7→ u∗(., t) is analytic from R into Xα with α ∈ (1/2, 1). In particular, for all x ∈ Ω,u∗(x, t) is analytic with respect to the time.
Proof: Theorem 5.1 uses strongly some compactness properties. Therefore, we need totruncate our solution to apply the theorem on a bounded domain (of course, this is notnecessary and easier if Ω is already bounded).
Let χ ∈ C∞0 (Ω) be such that ∂χ
∂ν= 0 on ∂Ω, χ ≡ 1 in x ∈ Ω, γ(x) = 0 and supp(χ)
is included in a smooth bounded subdomain O of Ω. Since Proposition 4.4 shows thatu∗ ∈ C0(R, D(A)) and since u∗ is constant with respect to the time in supp(γ), (1 − χ)u∗
is obviously analytic from R into D(A). It remains to obtain the analyticity of χu∗.In this proof, the damping γ needs to be more regular than just L∞(Ω). We replace
γ by a damping γ ∈ C∞(Ω), which has the same geometrical properties (GCC) and (1.3)and which vanishes where γ does. Notice that γ∂tu
∗ ≡ 0 ≡ γ∂tu∗, therefore replacing γ by
γ has no consequences here.Let v = χu∗, we have
∂2ttv + γ(x)∂tv = ∆v − βv + g(x, v) (x, t) ∈ O × R+ ,
v(x, t) = 0 (x, t) ∈ ∂O × R+(5.2)
with g(x, v) = −χ(x)f(v+(1−χ)u∗(x))−2(∇χ.∇u∗)(x)− (u∗∆χ)(x). We apply Theorem5.1 with the following setting. Let Y = Xα = H1+α(O)∩H1
0 (O)×Hα0 (O) with α ∈ (1/2, 1).
Let V = (v, ∂tv) and let G(v) = (0, g(., v)). We set
A = A1 +B1 =
(0 Id
∆− β 0
)+
(0 00 −γ
).
Let (λk)k≥1 be the negative eigenvalues of the Laplacian operator on O with Dirichletboundary conditions and let (ϕk) be corresponding eigenfunctions. We set Pn to be thecanonical projections of X on the subspace generated by ((ϕk, 0))k=1...n and ((0, ϕk))k=1...n.
24
To finish the proof of Proposition 5.2, we only have to check that the hypotheses ofTheorem 5.1 hold.
The trajectory V is compact since we know by Proposition 4.4 that it is bounded inX1, which gives (i).
Hypothesis (ii) and (iii) hold with K0 = 1 by construction of Pn and because B1 isbounded in Y since γ belongs to C∞(Ω).
The first part of Hypothesis (iv) follows from Proposition 2.3. The second estimate‖e(A1+QnB1)t‖L(QnY,Y ) ≤ Me−λt means that the restriction QnAQn of A to the high fre-quencies of the wave operator also generates a semigroup satisfying an exponential decay.By a result of Haraux in [29], we know that this also holds, see Section 2.3.2 of [25] for thedetailed arguments.
We recall that u∗(x, .) is constant outside χ−1(1) and belongs locally to H1+α sinceu∗ ∈ D(A). Therefore, the terms (1 − χ)u∗(x), ∇χ.∇u∗ and u∗∆χ appearing in thedefinition of g are in H1. Moreover, they satisfy Dirichlet boundary condition on ∂Ω sinceu∗ ≡ 0 and ∂νχ ≡ 0 there. Of course, they also satisfy Dirichlet boundary condition on theother parts of ∂O since χ ≡ 0 outside O. Notice that α > 1/2 and thus H1+α(O)∩H1
0 (O)is an algebra included in C0. Therefore, (1.2) shows that G is of class C1 in the boundedsets of Y . Since u ∈ [−4KM, 4KM ] 7→ f(u) ∈ R is analytic, it can be extended toa holomorphic function in [−4KM, 4KM ] + i[−ρ, ρ] for small ρ > 0. Using again theembedding H1+α(O) → C0(O) and the definitions of K and M , we deduce that (v) holds.
Finally, for V2 = (v2, ∂tv2) with ‖V2‖Y ≤ 1 DG(V (t))V2 = (0,−χ(x)f ′(v(t) + (1 −χ)u∗(x))v2) is relatively compact in Y since v(t) is bounded in H2∩H1
0 due to Proposition4.4 and therefore v2 ∈ H1+α 7→ χ(x)f ′(v(t) + (1 − χ)u∗(x))v2 ∈ Hα is a compact map.This yields (vi).
Once the time-regularity of u∗ is proved, the space-regularity follows directly.
Proposition 5.3. Let f and u∗ be as in Proposition 5.2, then u∗ ∈ C∞(Ω× R).
Proof: Proposition 5.2 shows that u∗ and all its time-derivatives belongs to Xα withα ∈ (1/2, 1). Due to the Sobolev embeddings, this implies that any time-derivative of u∗
is Holder continuous. Writing
∆u∗ = ∂2ttu
∗ + βu∗ + f(u∗) (5.3)
and using the local elliptic regularity properties (see [42] and the references therein forexample), we get that u∗ is locally of class C2,λ in space for some λ ∈ (0, 1). Thus, u∗ isof class C2,λ in both time and space. Then, we can use a bootstrap argument in (5.3) toshow that u∗ is of class C2k,λ for all k ∈ N.
25
5.2 Identification of u∗
The smoothness and the partial analyticity of u∗ shown in Propositions 5.2 and 5.3 enableus to use the unique continuation result of [47].
Proposition 5.4. Let f and u∗ be as in Proposition 5.2, then u∗ is constant in time, i.e.u∗ is an equilibrium point of the damped wave equation (1.1).
Proof: Setting v = ∂tu∗, we get ∂2
ttv = ∆v − βv− f ′(u∗)v. Propositions 5.2 and 5.3 showthat u∗ is smooth and analytic with respect to the time and moreover v ≡ 0 in supp(γ).Thus, the unique continuation result stated in Corollary 3.2 yields v ≡ 0 everywhere.
The sign assumption on f directly implies that 0 is the only possible equilibrium pointof (1.1).
Corollary 5.5. Let f ∈ C1(R) satisfying (1.2) and let E0 ≥ 0. Let K and M be theconstants given by Propositions 4.4 and 4.5 and assume that f is analytic in [−4KM, 4KM ].Then, the unique solution u∗ of (5.1) with E(u∗) ≤ E0 is u∗ ≡ 0.
Proof: Due to Proposition 5.4, u∗ is solution of ∆u∗ − βu∗ = f(u∗). By multiplying byu∗ and integrating by part, we obtain
∫Ω|∇u∗|2 + β|u∗|2 dx = −
∫Ωu∗f(u∗) dx, which is
non-positive due to Assumption (1.2). Since β ≥ 0 is such that ∆− β is negative definite,this shows that u∗ ≡ 0.
6 Proof of Theorem 1.1
Due to Proposition 2.5, Theorem 1.1 directly follows from the following result.
Proposition 6.1. Let f ∈ C1(R) satisfying (1.2) and let E0 ≥ 0. Let K and M be theconstants given by Propositions 4.4 and 4.5. Assume that f is analytic in [−4KM, 4KM ]and that γ is as in Theorem 1.1. Then, there exist T > 0 and C > 0 such that for any usolution of (1.1) with E(u)(0) ≤ E0 satisfies
E(u)(0) ≤ C
∫∫
[0,T ]×Ω
γ(x) |∂tu|2 dtdx.
Proof: We argue by contradiction: we assume that there exists a sequence (un) of solutionsof (1.1) and a sequence of times (Tn) converging to +∞ such that
∫∫
[0,Tn]×Ω
γ(x) |∂tun|2 dtdx ≤ 1
nE(un)(0) ≤
1
nE0. (6.1)
26
Denote αn = (E(un)(0))1/2. Since α ∈ [0,
√E0], we can assume that αn converges to a
limit α when n goes to +∞. We distinguish two cases: α > 0 and α = 0.
• First case: αn −→ α > 0Notice that, due to (2.1), ‖(un, ∂tun)(0)‖X is uniformly bounded from above and frombelow by positive numbers. We set u∗
n = un(Tn/2+ .). Due to the asymptotic compactnessproperty stated in Proposition 4.3, we can assume that u∗
n converges to a solution u∗ of(1.1) in C0([−T, T ], X) for all time T > 0. We notice that
E(un(0)) ≥ E(u∗n(0)) = E(un(0))−
∫∫
[0,Tn/2]×Ω
γ(x) |∂tun|2 ≥ (1− 1/n)E(un(0))
and thus E(u∗(0)) = α2 > 0. Moreover, (6.1) shows that γ∂tu∗n converges to zero in
L2([−T, T ], L2(Ω)) for any T > 0 and thus ∂tu∗ ≡ 0 in supp(γ). In other words, u∗ is a
non-dissipative solution of (1.1), i.e. a solution of (5.1) with E(u∗) = α2 ≤ E0. Corollary5.5 shows that u∗ ≡ 0, which contradicts the positivity of E(u∗(0)).
• Second case: αn −→ 0The assumptions on f allow to write f(s) = f ′(0)s+R(s) with
|R(s)| ≤ C(|s|2 + |s|p) and |R′(s)| ≤ C(|s|+ |s|p−1) . (6.2)
Let us make the change of unknown wn = un/αn. Then, wn solves
wn + γ(x)∂twn + (β + f ′(0))wn +1
αn
R(αnwn) = 0 (6.3)
and ∫∫
[0,Tn]×Ω
γ(x) |∂twn|2 dtdx ≤ 1
n. (6.4)
Denote Wn = (wn, ∂twn). Due to the equivalence between norm and energy given by (2.1),the scaling wn = un/αn implies that ‖(wn(0), ∂twn(0))‖X is uniformly bounded from aboveand from below by positive numbers. Moreover, (6.1) implies
‖Wn(t)‖X =‖(Un(t))‖X
αn≥ C
E(un(t))1/2
αn≥ C
(E(un)(0)− α2n/n)
1/2
αn≥ C
2> 0 (6.5)
for any t ∈ [0, Tn] and n large enough.We set fn = 1/αnR(un) and Fn = (0, fn). The stability estimate of Proposition 2.2
implies that ‖un‖Lq([k,k+1],Lr) ≤ Cαn uniformly for n, k ∈ N. In particular, combined with(6.2), this gives
‖fn‖L1([k,k+1],L2) =
∥∥∥∥1
αnR(αnwn)
∥∥∥∥L1([k,k+1],L2)
≤ C(αn + αp−1n )
27
We can argue as in Proposition 4.3 and write
Wn(Tn) = eATnWn(0) +
⌊Tn⌋−1∑
k=0
eA(Tn−k)
∫ 1
0
e−AsFn(k + s) ds
+ eA(Tn−⌊Tn⌋)
∫ Tn−⌊Tn⌋
0
e−AsFn(⌊Tn⌋ + s) ds . (6.6)
where A is the modified damped wave operator
A =
(0 Id
∆− β − f ′(0) −γ
).
Notice that A satisfies an exponential decay as A in Proposition 2.3 since (1.2) impliesf ′(0) ≥ 0. By summing up as in Proposition 4.3, we get
‖Wn(Tn)‖X ≤ Ce−λTn + C(αn + αp−1n )
which goes to zero, in a contradiction with (6.5).
As a direct consequence of Proposition 6.1, we obtain a unique continuation propertyfor nonlinear wave equations. Notice that the time of observation T required for theunique continuation is not explicit. Thus, this result is not so convenient as a uniquecontinuation property. But it may be useful for other nonlinear stabilization problems asu+ γ(x)g(∂tu) + f(u) = 0.
Corollary 6.2. Let f ∈ C1(R) satisfying (1.2) and let E0 ≥ 0. Assume that f is analyticin R and that ω is an open subset of Ω satisfying (GCC). Then, there exist T > 0 suchthat the only solution u of
u+ βu+ f(u) = 0 on [−T, T ]× Ω
∂tu ≡ 0 on [−T, T ]× ω .(6.7)
with E(u)(0) ≤ E0 is u ≡ 0.
Proof: Corollary 6.2 is a straightforward consequence of Proposition 6.1 since we can easilyconstruct a smooth damping γ supported in ω and such that Supp(γ) satisfies (GCC). Weonly have to remark that u solution of (6.7) is also solution of (1.1).
28
7 Proof of Theorem 1.2
Before starting the proof of Theorem 1.2 itself, we prove that C1(R) is a Baire space,
that is that any countable intersection of open dense sets is dense. This legitimizes thegenericity in C
1(R) as a good notion of large subsets of C1(R). We recall that C1(R) is
defined by (1.6) and endowed by Whitney topology, the open sets of which are generatedby the neighbourhoods Nf,δ defined by (1.7).
Proposition 7.1. The space C1(R) endowed with Whitney topology is a Baire space.
Proof: The set C1(R) is not an open set of C1(R), and neither a submanifold. It is a
closed subset of C1(R), but C1(R) endowed with Whitney topology is not a completelymetrizable space, since it is not even metrizable (the neighbourhoods of a function f arenot generated by a countable subset of them). Therefore, we have to come back to thebasic proof of Baire property as in [23].
Let U be an open set of C1(R) and let (On)n∈N be a sequence of open dense sets ofC1(R). By density, there exists a function f0 ∈ C
1(R) in U ∩ O0 and by openness, thereexists a positive continuous function δ0 such that the neighbourhood Nf0,δ0 is containedin U ∩ O0. By choosing δ0 small enough, one can also assume that Nf0,2δ0 ⊂ U ∩ O0 andthat supu∈R |δ0(u)| ≤ 1/20. By recursion, one constructs similar balls Nfn,δn ⊂ Nfn−1,δn−1
⊂Nf0,δ0 such that Nfn,2δn ⊂ U∩On and that supu∈R |δn(u)| ≤ 1/2n. Since C1([−m,m],R) en-dowed with the uniform convergence topology is a complete metric space, the sequence (fi)converges to a function f ∈ C1(R,R) uniformly in any compact set of R. By construction,the limit f satisfies
∀n ∈ N, ∀u ∈ R , max(|f(u)− fn(u)|, |f ′(u)− f ′n(u)|) ≤ δn(u) < 2δn(u) , (7.1)
as well as f(0) = 0 and uf(u) ≥ 0 since any fn satisfies (1.2). Moreover, there exist C > 0and p ∈ [1, 5) such that f0 satisfies
|f0(u)| ≤ C(1 + |u|)p and |f ′0(u)| ≤ C(1 + |u|)p−1 . (7.2)
Since max(|f(u)− f0(u)|, |f ′(u)− f ′0(u)|) ≤ δ0(u) ≤ 1, f also satisfies (7.2) with a constant
C ′ = C+1. Therefore, f satisfies (1.2) and thus belongs to C1(R). In addition, f satisfying
(7.1) and Nfn,2δn being contained in U ∩On, we get f ∈ U ∩On for all n. This shows that∩n∈NOn intersects any open set U and therefore is dense in C
1(R).
Proof of Theorem 1.2: We denote by Gn the set of functions f ∈ C1(R) such that the
exponential decay property (ED) holds for E0 = n. Obviously, G = ∩n∈NGn and hence itis sufficient to prove that Gn is an open dense subset of C1(R). We sketch here the mainarguments to prove this last property.
29
Gn is a dense subset: letN be a neighbourhood of f0 ∈ C1(R). Up to choosingN smaller,
we can assume that the constant in (1.2) is independent of f ∈ N . Due to Propositions4.4 and 4.5, there exist constants K and M such that, for all f ∈ N , all the global non-dissipative trajectories u of (1.1) with E(u) ≤ n are such that ‖u‖L∞(Ω×R) ≤ KM . Weclaim that we can choose f ∈ N as close to f0 as wanted such that f is analytic on[−4KM, 4KM ] and still satisfies (1.2). Then, Proposition 6.1 shows that f satisfies (ED)with E0 = n i.e. that f ∈ Gn.
To obtain this suitable function f , we proceed as follows. First, we set a = 4KM andnotice that it is sufficient to explain how we construct f in [−a, a]. Indeed, one can easilyextend a perturbation f of f0 in [−a, a] satisfying f(s)s ≥ 0 to a perturbation f of f0 inR, equal to f0 outside of [−a − 1, a + 1] and such that f(s)s ≥ 0 in [−a − 1, a + 1]. Weconstruct f in [−a, a] as follows. Since f0(s)s ≥ 0, we have that f ′
0(0) ≥ 0. We perturbf0 to f1 such that f1(0) = 0, f ′
1(s) ≥ ε > 0 in a small interval [−η, η] and sf1(s) ≥ 2ε in[−a,−η]∪ [η, a], where ε could be chosen as small as needed. Then we perturb f1 to obtaina function f2 which is analytic in [−a, a] and satisfies f ′
2(s) > 0 in [−η, η], sf2(s) ≥ ε in[−a,−η]∪ [η, a] and |f2(0)| < ε/a. Finally, we set f(s) = f2(s)− f2(0) and check that f isanalytic and satisfies sf(s) ≥ 0 in [−a, a]. Moreover, up to choosing ε very small, f is asclose to f0 as wanted.
Gn is an open subset: let f0 ∈ Gn. Proposition 2.3 shows the existence of a constantC and a time T such that, for all solution u of (1.1),
E(u(0)) ≤ E0 =⇒ E(u(0)) ≤ C
∫ T
0
∫
Ω
γ(x)|∂tu(x, t)|2 dxdt . (7.3)
The continuity of the trajectories in X with respect to f ∈ C1(R) is not difficult to obtain:
using the strong control of f given by Whitney topology, the arguments are the same asthe ones of the proof of the continuity with respect to the initial data, stated in Theorem2.2. Thus, (7.3) holds also for any f in a neighbourhood N of f0, replacing the constantC by a larger one. Therefore, Proposition 2.3 shows that N ⊂ Gn and hence that Gn isopen.
8 A proof of compactness and regularity with the
usual arguments of control theory
In this section, we give an alternative proof of the compactness and regularity properties ofPropositions 4.3 and 4.4. We only give its outline since it is redundant with previous resultsof the article. Moreover, it is quite similar to the arguments of [17]. Yet, the arguments
30
of this section are interesting because they do not require any asymptotic arguments andthey show a regularization effect through an observability estimate with a finite time T ,which can be explicit. However, for the moment, it seems impossible to obtain an analyticregularity similar to Proposition 5.2 with this kind of arguments.
Instead of using a Duhamel formula with an infinite interval of time (−∞, t) as in (4.3),the main idea is to use as a black-box an observability estimate for T large enough, T beingthe time of geometric control condition,
‖U0‖2Xs ≤ C∥∥BetAU0
∥∥2
L2([0,T ],Xs)(8.1)
where
A =
(0 Id
∆− β −γ
)and B =
(0 00 −γ
).
The first aim is to prove that a solution of (5.1), globally bounded in energy, is alsoglobally bounded in Xs for s ∈ [0, 1]. We proceed step by step. First, let us show that itis bounded in Xε.
• we fix T > large enough to get the observability estimate (8.1). By the existencetheory on each [t0, t0 + T ], u|[t0,t0+T ] is bounded in Strichartz norms, uniformly fort0 ∈ R. Since the nonlinearity is subcritical, Corollary 4.2 gives that f(u) is globallybounded in L1([t0, t0 + T ], H1+ε).
• we decompose the solution into its linear and nonlinear part by the Duhamel formula
U(t) = eA(t−t0)U(t0) +
∫ t
t0
eA(t0−τ)f(U(τ))dτ = Ulin + UNlin.
Since f(u) is bounded in L1([t0, t0+T ], H1+ε), UNlin is uniformly bounded in C([t0, t0+T ], Xε).
• we will now use the linear observability estimate (8.1) with s = ε, applying it to Ulin:
‖U(t0)‖2Xε = ‖Ulin(t0)‖2Xε ≤ C
∫ t0+T
t0
‖γ(x)∂tulin‖2Hε . (8.2)
Then, using triangular inequality, we get∫ t0+T
t0
‖γ(x)∂tulin‖2Hε ≤ 2
∫ t0+T
t0
‖γ(x)∂tu‖2Hε + 2
∫ t0+T
t0
‖γ(x)∂tuNlin‖2Hε
≤ 2
∫ t0+T
t0
‖γ(x)∂tuNlin‖2Hε ≤ C
where we have used ∂tu ≡ 0 on ω and UNlin is bounded in C([t0, t0 + T ], Xε). Com-bining this with (8.2) for any t0 ∈ R, we obtain that U is uniformly bounded in Xε
on R.
31
Repeating the arguments, we show that u is bounded in X2ε, X3ε and so on. . . until X1.Similar ideas allows to prove some theorem of propagation of compactness in finite time,replacing the asymptotic compactness property of Proposition 4.3.
As said above, an advantage of this method, compared to the one used in Propositions4.3 and 4.4, is that it allows to propagate the regularity or the compactness on some finiteinterval of fixed length. Yet, it seems that such propagation results are not available in theanalytic setting. Indeed, it seems that, for nonlinear equations, the propagation of analyticregularity or of nullity in finite time is much harder to prove. We can for instance referto the weaker (with respect to the geometry) result of Alinhac-Metivier [1] or the negativeresult of Metivier [40].
9 Applications
9.1 Control of the nonlinear wave equation
In this subsection, we give a short proof of Theorem 1.3, which states the global control-lability of the nonlinear wave equation. The first step consists in a local control theorem.
Theorem 9.1 (Local control). Let ω satisfying the geometric control condition for a timeT . Then, there exists δ such that for any (u0, u1) in H1
0 (Ω)× L2(Ω), with
‖(u0, u1)‖H10×L2 ≤ δ
there exists g ∈ L∞([0, T ], L2) supported in [0, T ]× ω such that the unique strong solutionof
u+ βu+ f(u) = g on [0, T ]× Ω
(u(0), ∂tu(0)) = (u0, u1).
satisfies (u(T ), ∂tu(T )) = (0, 0).
Proof: The proof is exactly the same as Theorem 3 of [17] or Theorem 3.2 of [38]. Themain argument consists in seeing the problem as a perturbation of the linear controllabilitywhich is known to be true in our setting.
Now, as it is very classical, we can combine the local controllability with our stabiliza-tion theorem to get global controllability.
Sketch of the proof of Theorem 1.3: In a first step, we choose as a control g =−γ(x)∂tu where u is solution of (1.1) with initial data (u0, u1). By uniqueness of solutions,we have u = u. Therefore, thanks to Theorem 1.1, for a large time T1, only depending on
32
R0, we have ‖(u(T1), ∂tu(T1))‖H1×L2 ≤ δ. Then, Theorem 9.1 allows to find a control thatbrings (u(T1), ∂tu(T1)) to 0. In other words, we have found a control g supported in ω thatbrings (u0, u1) to 0. We obtain the same result for (u0, u1) and conclude, by reversibilityof the equation, that we can also bring 0 to (u0, u1).
9.2 Existence of a compact global attractor
In this subsection, we give the modification of the proofs of this paper necessary to getTheorem 1.4 about the existence of a global attractor.
The energy associated to (1.9) in X = H10 (Ω)× L2(Ω) is given by
E(u, v) =
∫
Ω
1
2(|∇u|2 + |v|2) + V (x, u)dx ,
where V (x, u) =∫ u
0f(x, ξ)dξ.
The existence of a compact global attractor for (1.9) is well known for the Sobolevsubcritical case p < 3. The first proofs in this case go back to 1985 ([24] and [27]), see[45] for other references. The case p = 3 as been studied in [5] and [4]. For p ∈ (3, 5),Kapitanski proved in [34] the existence of a compact global attractor for (1.9) if Ω is acompact manifold without boundary and if γ(x) = γ is a constant damping. Using thesame arguments as in the proof of our main result, we can partially deal with the casep ∈ (3, 5) with a localized damping γ(x) and with unbounded manifold with boundaries.
Assume that f satisfies the assumptions of Theorem 1.4. We first claim that we canassume in addition that f(x, 0) = 0 on ∂Ω in order to guarantee the Dirichlet boundarycondition for f(x, u) if u ∈ H1
0 (Ω). Indeed, let ϕ be the solution of ∆ϕ−βϕ = f(x, 0) withDirichlet boundary conditions, which is well defined and smooth, since by assumptionsf(x, 0) is smooth and compactly supported. Let χ be a smooth compactly supported cut-off function such that ∂χ
∂ν= 0 on ∂Ω and χ ≡ 1 in the ball B(x0, R) of Hypothesis (1.11).
If f(x, 0) 6= 0 on ∂Ω, we consider the new variable u = u− χϕ and the new equation
∂2ttu+ γ(x)∂tu = ∆u− βu− f(x, u) (9.1)
where f(x, u) = f(x, u+ χϕ(x))− χf(x, 0) + 2∇ϕ.∇χ+ ϕ∆χ. One directly checks that usolves (1.9) if and only if u solves (9.1). Moreover f is smooth, analytic in u and satisfiesHypotheses (1.10) and (1.11) and in addition f(x, 0) = 0 on the boundary. Clearly, sinceour change of variables is a simple translation, obtaining a compact global attractor for(9.1) is equivalent to obtaining a compact global attractor for (1.9). In what follows, wemay thus assume that f(x, 0) = 0 in addition to the hypotheses of Theorem 1.4.
The arguments of this paper show the following properties.
33
i) The positive trajectories of bounded sets are bounded. Indeed, (1.11) impliesthat, for x 6∈ B(x0, R), we have V (x, u) =
∫ u
0f(x, ξ)dξ ≥ 0. Moreover, for x ∈ B(x0, R),
V (x, .) is non-increasing on (−∞,−R) and non-decreasing on (R,∞). Thus, V (x, u) isbounded from below for x ∈ B(x0, R) and
∀(u, v) ∈ X , E(u, v) ≥ 1
2‖(u, v)‖2X + vol(B(x0, R)) inf V .
The Sobolev embeddings H1(Ω) → Lp+1(Ω) shows that the bounded sets of X have abounded energy. Since the energy E is non increasing along the trajectories of (1.9), weget that the trajectory of a bounded set is bounded.
ii) The dynamical system is asymptotically smooth. The asymptotic compactnessexactly corresponds to the statement of Proposition 4.3. Let us briefly explain why itcan be extended to the case where f depends on x. The key point is the extension ofCorollary 4.2. First notice that, since we have assumed that f(x, 0) = 0 on ∂Ω, f(x, u)satisfies Dirichlet boundary condition if u does. Then, it is not difficult to see thatthe discussion following Theorem 4.1 can be extended to the case f depending on x byusing estimates (1.10). Corollary 4.2 follows then, except for a small change: since itis possible that f(x, 0) 6= 0 for some x ∈ Ω, the conclusion of Corollary 4.2 should bereplaced by
‖f(x, v)‖L1([0,T ],Hs+ε0
(Ω)) ≤ C(1 + ‖v‖L∞([0,T ],Hs+1(Ω)∩H10(Ω))) .
Then the proof of Proposition 4.3 is based on Corollary 4.2, the boundedness of thepositive trajectories of bounded sets (both could be extended to the case where f de-pends on x as noticed above) and an application of Proposition 2.6 outside of a largeball. We conclude by noticing that, for x large, f(x, u)u ≥ 0 and γ(x) ≥ α > 0 andthus Proposition 2.6 can still be applied exactly as in the proof of Proposition 4.3.
iii) The dynamical system generated by (1.9) is gradient, that is that the energy E isnon-increasing in time and is constant on a trajectory u if and only if u is an equilibriumpoint of (1.9). This last property is shown in Proposition 5.4 for f independent of xbut can be easily generalised for f = f(x, u). Notice that the proof of this property isthe one, where the analyticity of f is required since the unique continuation property ofSection 3 is used. Finally, we enhance that the gradient structure of (1.9) is interestingfrom the dynamical point of view since it implies that any trajectory u(t) convergeswhen t goes to +∞ to the set of equilibrium points.
iv) The set of equilibrium points is bounded. The argument is similar to the one ofCorollary 5.5: if e is an equilibrium point of (1.9) then (1.11) implies that
∫1
2|∇e|2 + β|e|2 = −
∫
Ω
f(x, e)edx ≤ −vol(B(x0, R)) inf(x,u)∈Ω×R
f(x, u)u ,
where we have bounded f(x, u)u from below exactly as we have done for V (x, u) in i).
34
It is well known (see [24] or Theorem 4.6 of [45] for examples) that Properties i)-iv) yieldthe existence of a compact global attractor. Hence, we obtain the conclusion of Theorem1.4.
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